A semilinear elliptic equation

We study solutions of the semilinear elliptic partial differential equation Δ u = $1u$ on domains in $Rn$. Our main goal is to prove study the singular solutions, i.e. non-negative functions which are limits of positive smooth solutions. The equation Δ u = $1u$ is the Euler-Lagrange equation for the functional . We are able to prove the existence of a large variety of smooth positive solutions which are stable in the sense that the second variation of $F$ is nonnegative. There are two main results for stable solutions. The first gives a Hölder continuity bound for stable solutions on compact subdomains. The second shows that in dimensions 2 ≤ n ≤ 6, stable solutions are bounded away from zero, and thus cannot converge to singular solutions.